Master Thesis Energy Harvesting Receiver Modelling and Resource Allocation in SWIPT Elena Boshkovska Lehrstuhl für Digitale Übertragung Prof. Dr.-Ing. Robert Schober Universität Erlangen-Nürnberg Supervisor: Dr. Derrick Wing Kwan Ng Prof. Dr.-Ing. Robert Schober September 17, 2015
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Master Thesis
Energy Harvesting Receiver Modelling andResource Allocation in SWIPT
Elena Boshkovska
Lehrstuhl für Digitale ÜbertragungProf. Dr.-Ing. Robert Schober
Universität Erlangen-Nürnberg
Supervisor: Dr. Derrick Wing Kwan NgProf. Dr.-Ing. Robert Schober
Simultaneous wireless information and power transfer (SWIPT) is a promising solution
for enabling long-life, and self-sustainable wireless networks. In this thesis, we propose
a practical non-linear energy harvesting (EH) model and design a resource allocation
algorithm for SWIPT systems. In particular, the algorithm design is formulated as a
non-convex optimization problem for the maximization of the total harvested power
at the EH receivers subject to quality of service (QoS) constraints for the information
decoding (ID) receivers. To circumvent the non-convexity of the problem, we transform
the corresponding non-convex sum-of-ratios objective function into an equivalent objec-
tive function in parametric subtractive form. Furthermore, we design a computationally
efficient iterative resource allocation algorithm to obtain the globally optimal solution.
Numerical results illustrate significant performance gain in terms of average total har-
vested power for the proposed non-linear EH receiver model, when compared to the
traditional linear model.
Publications related to the thesis:
• E. Boshkovska, D. W. K. Ng, N. Zlatanov, and R. Schober, “Practical Non-linear
Energy Harvesting Model and Resource Allocation for SWIPT Systems," IEEE
Commun. Lett., vol. 19, pp. 2082 - 2085, Dec. 2015.
• E. Boshkovska, R. Morsi, D. W. K. Ng, and R. Schober, “Power Allocation and
Scheduling for SWIPT Systems with Non-linear Energy Harvesting Model," ac-
cepted for presentation at IEEE ICC 2016.
• E. Boshkovska, D. W. K. Ng, N. Zlatanov, and R. Schober, “Robust Beamforming
for SWIPT systems with Non-linear Energy Harvesting Model," invited paper,
17th IEEE International workshop on Signal Processing advances in Wireless
Communications, Jul. 2016.
vi
vii
Glossary
Abbreviations
AC Alternating CurrentAWGN Additive White Gaussian NoiseCMOS Complementary Metal-oxide-semiconductorCSI Channel State InformationDC Direct CurrentEH Energy HarvestingER Energy Harvesting ReceiverFDD Frequency Division DuplexGSM Global System for Mobile CommunicationsID Information DecodingISM Industrial, Scientific, and MedicalMISO Multiple-Input Single-OutputMIMO Multiple-Input Multiple-OutputNP-hard Non-deterministic Polynomial-time hardQoS Quality of ServiceRF Radio FrequencySINR Signal-to-interference-plus-noise RatioSWIPT Simultaneous Wireless Information and Power TransferTDD Time Division DuplexWIPT Wireless Information and Power TransferWPN Wireless Powered NetworkWPT Wireless Power Transfer
B BandwidthK Number or users in the systemT Number of time slotsPav Average radiated powerPmax Maximum radiated power at one time slotη RF-to-DC power conversion efficiency
Ever since wireless networks have been deployed in our surroundings, there has been an
exponential growth of the data rate requirements that these networks need to satisfy
along with the increasing demand for new and improved services. Under this content,
several significant technologies, such as multiple-input multiple-output (MIMO), capacity
achieving codes, and small-cell networks, have been proposed to tremendously increase
the speeds in wireless networks [4]. However, the demands in high quality of service
(QoS) increase the amount of required energy that wireless networks need to operate,
in both the transmitters to the end users, i.e., the mobile devices. The bottleneck that
slows down the evolution of communication networks is mainly at the mobile devices,
due to their limited energy supply. In particular, the development of battery capacity
has not been keeping up with the evolution of other network constituents. In the last
decade, extensive research has been conducted to study alternative solutions that might
offer ways to surpass the limitations caused by batteries. An appealing solution is energy
harvesting (EH), which has become very popular in the field of communications for
enabling self-sustainable mobile devices [5]. With the intelligence to harvest energy
from different sources, such as solar and wind, the lifetime of communication networks
can be increased along with enabling self-sustainability at the mobile terminals. However,
these natural sources have limited availability and are usually constrained by weather
and geographical location. One of the possible solutions to go beyond these limitations
is via the concept of wireless power transfer (WPT), which was first introduced in Tesla’s
work [6], published in the early 20th century. Yet, researchers started investigating the
possibilities of using WPT for charging end-user devices in wireless networks decades
later [7]. The opportunity rose due to the rapid advancement of microwave technologies
in the 1960s, along with the invention of rectifying antennas. The energy in WPT can be
harvested from either ambient radio frequency (RF) signals, or in a dedicated manner
from more powerful energy sources, e.g. base stations [5]. In the last decades, due
to the increasing number of wireless communication devices and sensors, the focus
2 Chapter 1. Introduction
has been set on recycling power from an omnipresent source of energy, i.e., harvesting
power1 from the energy of RF signals. Recently, simultaneous wireless information
and power transfer (SWIPT) has drawn much attention in the research community
[8]–[10]. In order to unify the transmission of information along with the process of
EH, the receivers in SWIPT systems reuse the energy that the RF signals carry in order to
supply the batteries of mobile devices while decodes the information successfully. In the
case when certain users are the energy harvesters and others are information receivers,
the concept is referred to as wireless information and power transfer (WIPT). Another
similar emerging concept is the wireless powered network (WPN) [11, 12, 13], where
the receivers rely solely on the power harvested from the appointed transmitter and use
that power for their future transmissions. In the following, we focus on SWIPT/WIPT
systems.
In this chapter, we give an overview of SWIPT, along with some specifics of receiver
modelling in SWIPT systems. Then, we state the motivation of the thesis.
1.1. Simultaneous Wireless Information and Power
Transfer
EH is a promising solution for overcoming the limitations introduced by energy-constrained
mobile devices. Moreover, when considering RF signals as an energy harvesting source,
we have an omnipresent, relatively stable, and controllable source of energy [14, 15].The harvested energy from the RF signals can be recycled and used as a supply to the
mobile devices in both indoor and outdoor environments. With existing EH circuits
available nowadays, we are able to harvest microwatts to milliwatts of power from
received RF signals over the range of several meters for a transmit power of 1 Watt
and a carrier frequency less than 1 GHz [16]. Thus, RF signals can be a viable energy
source for devices with low-power consumption, e.g. wireless sensor networks [17]. In
addition, we have the possibility to transmit energy along with the information signal
[18], which is known as SWIPT.
The receivers in a SWIPT system have the possibility to decode the transmitted
information and also harvest power that would be stored in their batteries for future
use. Ideally, the receivers in a SWIPT system would process the information at the same
time while harvesting energy from the same signal [14, 18]. However, due to practical
limitations, the EH receiver cannot reuse the power from the signal intended for decoding
in general. As a result, separate receivers that decouple the processes of information
1In this work, normalized unit energy is considered, i.e., Joule-per-second. In other words, the terms“energy" and “power" are interchangeable
1.1. Simultaneous Wireless Information and Power Transfer 3
decoding (ID) and EH using different policies have been presented in [19]–[30]. One
of the approaches for realizing this goal is implementing a power splitting receiver.
Specifically, the power splitting receiver splits the power of the incoming signal into
two power streams with power splitting ratios 1−ρ and ρ, for EH and ID, respectively.
The power splitting ratio 0 ≤ ρ ≤ 1 is previously determined for the power splitting
unit, which is installed at the analog front-end of the receiver. Power splitting receivers
in the context of SWIPT systems have been studied widely in the literature [31]–[35].Since we introduce the EH capability to the receiver side, a trade-off between ID and
EH arises naturally in such systems. Therefore, new resource allocation algorithms
that satisfy the requirements of SWIPT systems were investigated in [19]–[40]. The
fundamental trade-off between channel capacity and the amount of harvested energy,
considering a flat fading channel and frequency selective channel, was studied in
[14, 18, 19, 21]. Moreover, [36] and [37] focused on transmit beamforming design in
multiple-input single-output (MISO) SWIPT systems for separated and power splitting
receivers, respectively. Additionally, the concept of SWIPT was included in MIMO
system architectures in [41, 42]. The optimization of beamformers with the objective
to maximize the sum of total harvested energy under the minimum required signal-
to-interference-plus-noise ratio (SINR) constraints for multiple information receivers
was considered in [38]. In [22, 31, 33, 34, 42], resource allocation algorithms for
the maximization of the energy efficiency and spectral efficiency were developed in
different network architectures including SWIPT. These works have shown that the
energy efficiency can be improved by employing SWIPT in the considered communication
systems. In more recent works, [27, 43], the authors proposed multiuser scheduling
schemes, which exploit multiuser diversity for improving the system performance of
multiuser SWIPT systems. Besides, SWIPT has also been considered in cooperative
system scenarios [39, 44], where the performance of SWIPT systems is analysed by
considering different relaying protocols. Another aspect that is widely studied in the
literature is improving communication security in SWIPT systems [23]–[25], [30, 35].Namely, in order to facilitate EH at the receivers in SWIPT systems, the transmit power
is usually increased. Due to that fact, the susceptibility to eavesdropping might also be
increased. As a result, authors in [45]–[51] designed algorithms that provide physical
layer security in SWIPT systems. Furthermore, SWIPT has also been introduced in
cognitive networks [40, 48, 50], where cooperation between the primary and secondary
systems in a cognitive radio network with SWIPT was investigated. The abundance
of research demonstrated above implies that SWIPT leads to significant gains in many
aspects, for instance, energy consumption, spectral efficiency, and time delay. Therefore,
4 Chapter 1. Introduction
SWIPT is a novel concept that unlocks the potential of RF energy for developing self-
sustainable, long-life, and energy-efficient wireless networks.
1.2. Receiver Modelling in Wireless Information and
Power Transfer Systems
In this section, we focus on a widely adopted receiver model for EH in WIPT systems.
In general, a WIPT system consists of a transmitter of the RF signal, e.g. a base station,
that broadcasts the signal to the receivers, cf. Figure 1.1, as well as a receiver RF energy
harvesting node. After the signal has been received at the EH node, a chain of elements
process the signal as follows. The bandpass filter employed after the receiver antenna
performs the required impedance matching and passive filtering, before the RF signal
is passed to the rectifying circuit. The rectifier is a passive electronic device, usually
comprising diodes, resistors, and capacitors, that converts the incoming RF power to
direct current (DC) power, which can be stored in the battery storage of the receiver.
After the rectifier, usually a low-pass filter follows, in order to eliminate the harmonic
frequencies and prepare the power for storage.
RF EnergyHarvesing Node
BandpassFilter
Rectifying Circuit
Low-PassFilter
RFGenerator
Base Station
RF EH Circuit
Figure 1.1.: A point-to-point WPT system.
The end-to-end power conversion depends greatly on the characteristics of the rectify-
ing circuit. The rectifier can be implemented with different non-linear circuits, starting
from the simplest half wave rectifiers, cf. Figure 1.2, to complicated circuits that offer
N-fold increase of the circuit’s output and boost the efficiency of the circuit, cf. Figure
1.3. Half-wave rectifiers, as depicted in Figure 1.2, pass either the positive or negative
half of the alternating current (AC) wave, while the other half is blocked [52]. Even
though they result in a lower output voltage, the half-wave rectifiers comprise only one
diode in the simplest case. Thus, the half-wave rectifier is the simplest form of rectifier,
which is suitable for small mobile devices or wireless sensors. Figure 1.3 depicts an
array of voltage doubler circuits, where each part of the circuit consists of two diodes
1.2. Receiver Modelling in Wireless Information and Power Transfer Systems 5
D1
RF Cload Rload Vout+
-
Figure 1.2.: A schematic of a half-wave rectifier; Cload - Load capacitance, Rload - loadresistance, D1 - diode, and Vout - output voltage.
D1
D2
D3
D4C3
C1
C2
C4
RF
Cload
CN- 1 DN
DN- 1
Rload Vout
+
-
Figure 1.3.: A schematic of a Dickson charge pump; D1–DN diodes, and C1–CN capacitorsin stages 1–N .
and other corresponding elements. Depending on the number of stages required for a
particular rectifier, the circuit parts can be repeated until the N-th element is reached.
This configuration offers an increase of the conversion efficiency of the circuit, as well
as reducing the negative effects of a single circuit part. The rectifying circuits and
their optimization have been a research topic for decades [7], although recently more
attention has been drawn to them, due to their important role in WIPT/SWIPT systems
[53].
In [53], different configurations of rectifying circuits and an illustration of their
efficiency of converting the input RF power to output DC power have been presented.
On the other hand, the authors in [1] have developed a particular rectifier, suited
for the Global System for Mobile Communications (GSM) frequency band, which was
optimized to result in maximal conversion efficiency. Thus, the resulting configuration
that was built is a rectifying circuit with 36 stages in complementary metal-oxide
6 Chapter 1. Introduction
semiconductor (CMOS) technology. Another work in [2] analyzed a circuit configuration
that attempts to maximize the input before rectification by using a high-Q resonator
preceding the rectifier. Moreover, authors in [54] have studied three different techniques
for impedance matching and their influence on the efficiency of the rectifier. Design
of a dual-band rectifier for WIPT, whose efficiency is optimized in both the 2.4 GHz
and 5.8 GHz industrial, scientific and medical (ISM) bands, was presented in [55].Furthermore, the authors in [3] proposed a new method for analytical calculation of
the efficiency of microwave rectifiers. Most circuits shown in the literature comprise
different elements and are constructed in slightly different configurations. Despite the
abounding research in EH circuit design and their optimization, there is still no general
and tractable mathematical representation of the input-output response of a rectifying
circuit.
On the other hand, we expect that the input-output response of the EH circuit is
non-linear, considering that in any possible configuration, the rectifying circuit has at
least one non-linear element, such as the diode or diode-connected transistor. The
most important parameter that describes the capability of the rectifying circuit is the
RF-to-DC conversion efficiency. In general, the conversion efficiency is defined as the
ratio between the output DC power and the input RF power:
η=PDC-out
PRF-in, (1.2.1)
where PRF-in is the power of the RF signal that enters the rectifier and PDC-out is the
converted output DC power. The relationship that the efficiency described is shown
to be non-linear, due to the non-linear nature of the circuit itself. This non-linearity is
observed in all the measurements presented in [53]–[3], which were performed using
practical EH circuits. Similar non-linear behaviour also appears when we observe the
output DC power with respect to the input RF power, because they are also connected
through the conversion efficiency of the circuit. As aforementioned, the problem of
modelling the relationship between the input and output power of a rectifier through a
general expression has not been reported in the literature, yet. However, an accurate and
tractable model is necessary in order to include the effect of practical rectifying circuits
on the harvested power at the EH receivers, when working with SWIPT communication
systems.
In many recent works related to EH in communications, a specific linear model has
been assumed for describing the harvested power after the rectifying circuit [19]–[30].
1.3. Motivation 7
In particular, the output power is related to the input power through the conversion
efficiency η [19]:PDC-out = ηPRF-in. (1.2.2)
Furthermore, η is a constant that can take on values in the interval [0,1] and is
supposed to represent the capability of the RF-to-DC conversion circuit. The authors in
[19]–[30], as well as many others, assume the same model as in (1.2.2) for representing
the harvested power after the RF signal has been received and processed. Through
(1.2.2), a linear behaviour between the input and output power is introduced in the
system. With this model, the power conversion efficiency is independent of the input
power level of the EH circuit. In practice, the end-to-end wireless power transfer is
non-linear and is influenced by the parameters of the practical EH circuits, which are
built using at least one non-linear element, as it was previously shown. Thus, the
linear assumption for the conversion efficiency and for the EH receiver model does not
follow the actual characterization of practical EH circuits in general. More importantly,
significant performance losses may occur in SWIPT systems, when the design of resource
allocation algorithm is based on an inaccurate linear EH model.
1.3. Motivation
This thesis is motivated by the inaccuracy of the traditional linear EH receiver model
to capture the non-linear characteristic of the RF-to-DC power conversion in practical
RF EH systems. Specifically, the use of the conventional linear EH model may lead to
resource allocation mismatch in SWIPT systems, resulting in losses in the amount of
total harvested energy in the system.
In this thesis, we first focus on modelling a practical EH receiver circuit, which is
fundamentally important for the design of resource allocation algorithm in SWIPT
systems. To this end, an accurate and tractable EH model, which reflects the non-linear
nature of the practical EH circuit, is proposed. Alongside this model, we design a
resource allocation algorithm for the maximization of the total harvested power at the
EH receivers in the system, subject to QoS constraints. Furthermore, the proposed
practical non-linear model is compared to the existing linear EH model used in the
literature.
The rest of the thesis is organized in the following manner. In Chapter 2, we introduce
the communication system model adopted in the thesis. Afterwards, we propose a
practical non-linear EH model which is used in the resource allocation algorithm design.
Then, the results from the simulation framework are presented. Finally, we summarize
the contributions of this thesis in Chapter 3.
8
9
Chapter 2.
Resource Allocation Algorithm for aPractical EH Receiver Model
In this chapter, we focus on designing a resource allocation algorithm for a practical EH
receiver model in a SWIPT system. To this end, we first propose a practical non-linear EH
receiver model, which we adopt as an objective function for the design of the resource
allocation algorithm. We aim to maximize the average total harvested power at the
EH receivers in the system under some QoS constraints. The optimization problem is
formulated as a non-convex sum-of-ratios problem. After transforming the considered
non-convex objective function in sum-of-ratios form into an equivalent objective function
in parametric subtractive form, we present a computationally efficient iterative resource
allocation algorithm for achieving the globally optimal solution. At the end of the
chapter, numerical results for the underlying simulation framework are presented, where
the proposed EH receiver model is compared to the existing linear EH receiver model.
2.1. System Model
The system model for this work is depicted in Figure 2.1. We focus on a downlink
multiuser system, where a single-antenna base station broadcasts the RF signal to K
single-antenna users, which are capable of ID and EH. We assume that the users have
additional power supply, such that they do not rely solely on the RF EH for their battery
supply. Transmission in the system is divided into T unit time slots. For each time slot
n and each user k, we perform joint user selection and power allocation to optimize
the system performance. As for the channel model, we assume a frequency flat slow
fading channel. The channel impulse response is assumed to be time invariant during
each time slot n. Thus, the downlink channel state information (CSI) can be obtained by
exploiting feedback from users in frequency division duplex (FDD) systems and channel
reciprocity in time division duplex (TDD) systems. The base station excutes the resource
10 Chapter 2. Resource Allocation Algorithm for a Practical EH Receiver Model
Base station
User 1
User 2
User n
User K
Information flowPower flow
power to battery storage
Figure 2.1.: A multiuser SWIPT communication model.
allocation policy at each time slot n, based on the available CSI. Moreover, at each time
slot n the downlink received symbol at user k is given by
yk(n) =p
Pk(n)hk(n)xk(n) + zk(n), (2.1.1)
where xk(n) is the transmitted symbol, Pk(n) is the transmitter power, and hk(n) is the
channel gain coefficient describing the joint effects of multipath fading and path loss,
for user k at time slot n. For the transmitted symbol, we assume a zero mean symbol
with variance E|xk(n)|2= 1,∀n, k, where E· stands for statistical expectation. zk(n)represents the additive white Gaussian noises (AWGN) for time slot n and user k with
zero mean and equal variance σ2. Given perfect CSI at the user, the instantaneous
capacity for user k and time slot n is defined by
Ck(n) = log2
1+Pk(n)hk(n)
σ2
. (2.1.2)
At each time slot, only a single user is chosen to receive the information, i.e., to perform
ID, while the other K − 1 users can opportunistically harvest energy from the signal that
is radiated from the base station. Considering the fact that we focus on maximizing the
overall harvested power, in the following we focus only on the users selected for EH,
while also satisfying the QoS constraints for the ID users.
At the EH receiver, the users receive the signal through their antennas, which are
assumed to have ideal impedance matching. Then, the RF signal goes through the
rectification process, which converts the incoming RF power into output DC power. For
this part, instead of adopting the existing linear model for modelling the DC output
power, a non-linear conversion function for a practical EH receiver model is proposed.
2.2. Practical EH Receiver Model Proposition 11
The proposed power conversion function captures the effect of the practical rectifier on
the end-to-end RF-to-DC power conversion.
2.2. Practical EH Receiver Model Proposition
In this section, we propose a non-linear function that describes the input-output response
of a practical EH receiver. As it was previously elaborated in Chapter 1, the existing linear
model for the EH circuit does not capture the end-to-end non-linearity of a practical
EH receiver in a WIPT system and can lead to resource allocation mismatch for the
corresponding system. This can be avoided by adapting the model to the practical EH
circuits. For this purpose, we propose to use a logistic (sigmoidal) function, which is
a special kind of quasi-concave functions, to model the input-output characteristic of
the EH circuits. Its standard shape is shown in Figure 2.2, and the general analytical
expression has the following form:
f (x) =M
1+ e−a(x−b). (2.2.1)
In (2.2.1), M is the parameter that describes the value of the asymptote when input x →∞, a shows the steepness of the curve, and b is the mid-point of the curve. The logistic
−6 −4 −2 0 2 4 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 2.2.: A standard logistic function.
function can also take many different forms, with more or less parameters, depending
on the model and the specific application. It is used in many different fields of science,
for instance, modelling population growth, machine learning, as a utility function in
networking etc..
12 Chapter 2. Resource Allocation Algorithm for a Practical EH Receiver Model
To facilitate the development of a practical model for the end-to-end power conversion
in a practical EH circuit, we transform (2.2.1) into a slightly different form of the logistic
function:
PDC =M 1
1+e−a(PRF−b) −1
1+eab
1− 11+eab
. (2.2.2)
In (2.2.2), PDC is the output DC power, while PRF represents the RF power from the RF
signal that enters the rectifier, after the RF signal has been received and processed. We
note that equation (2.2.2) takes into account the zero-input/zero-output response of
EH circuits [56], which cannot be modelled by the function in (2.2.1). Constants M ,
a, and b in (2.2.2) describe the behaviour of the curve and comply with the general
definition of the parameters in the initial form of the function in (2.2.1). The value
of M is related to the maximum output DC power, i.e., the maximum power that can
be harvested through a particular circuit configuration. a and b show the steepness
and the inflexion point of the curve that describes the input-output power conversion.
Moreover, b is related to the minimum required turn-on voltage for the start of current
flow through the diode [53], while a reflects the non-linear charging rate with respect
to the input power. In general, these parameters depends on the choice of hardware
components for assembling the rectifier. Yet, once the EH circuit is fixed, the parameters
can be easily estimated through a curve fitting of the measurement data.
In the following, we verify the accuracy of the proposed model for the EH receiver by
reviewing the measurement results of rectifying circuits presented in [53]–[3]. More
specifically, we collect the data for the input versus output power and perform curve
fitting of the measurements. The results shown in Figures 2.3–2.5 were obtained using
the transformed form of the logistic function, defined in (2.2.2). The curve fitting was
done using the Curve Fitting Toolbox, available in MATLAB [57], with relatively high
accuracy. In particular, the average value of the curve fitting output parameter Adjusted
R-squared (Adjusted-R2) for the curves in Figures 2.3–2.5 is shown to be 0.9858. This
parameter is a special version of the coefficient of determination R2. In particular, the
ordinary R2 is a statistical measure of how close the data are to the fitted curve and
represents the ratio between explained variation and total variation of data. Adjusted-R2
is modified to include the number of observation and regression coefficients, in order to
result in a more accurate measure for the goodness of the fit. Both parameters result in
values from 0 to 1, where 1 indicates the best fit.
As mentioned before, the figures show the output DC power with respect to the input
RF power in Watts, for different power ranges corresponding to different rectifying
circuit architectures. It can be observed that the behaviour of the curves in the figures
is relatively similar. As the input power increases, the output power also starts to
2.2. Practical EH Receiver Model Proposition 13
0 1 2 3 4 5 6 7 8 9 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
Inputapowera[W]a*a10−4
Out
putap
ower
a[W]a*
a10−
a4
Curveafit
Measurementadata
Figure 2.3.: Curve fitting of measurement data from [1].
0 100 200 300 400 500 600 700 800 900 1000
0
5
10
15
20
25
30
35
40
45
50
Inputspowers[sµWs]
Out
putsp
ower
s[sµ
Ws]
Curvesfit
Measurementsdata
Figure 2.4.: Curve fitting of measurement data from [2].
14 Chapter 2. Resource Allocation Algorithm for a Practical EH Receiver Model
10 20 30 40 50 60 70
4
6
8
10
12
14
16
18
20
22
24
Inputdpowerd[mW]
Out
putdp
ower
d[mW
]
Curvedfit
Measurementddata
Figure 2.5.: Curve fitting of measurement data from [3].
increase after some point, known as the sensitivity of the rectifying circuit. After that,
the output power continues to increase until it reaches the saturation region. Due to the
limitations that take place in the rectifying circuit, the output DC power cannot surpass
this saturation value [53]. This limit value is one of the most important differences to the
linear model, where it was unrealistically assumed that the output DC power can linearly
increase arbitrarily, when the input RF power is increasing. Taking into consideration
that the behaviour of the power conversion curves shown in Figures 2.3–2.5 complies
with the trend that the logistic function in (2.2.2) follows, we propose the function in
(2.2.2) to model the input-output behaviour of a practical EH receiver. Moreover, the
logistic function is often exploited in many fields of science and its properties have been
investigated in depth in the literature.
In the next section, (2.2.2) is used as a building block of the optimization problem
formulation that follows, with the aim to design a resource allocation algorithm for the
system model presented above.
2.3. Resource Allocation Problem Formulation
The aim of the following section is the design of a jointly optimal power allocation and
user selection algorithm that maximizes the total harvested power for the considered
SWIPT system in Section 2.1. The objective is to maximize the total harvested power
at the EH receivers using the proposed practical EH receiver model. For this purpose,
2.3. Resource Allocation Problem Formulation 15
we adopt the power conversion function (2.2.2), which was modelled according to the
logistic function, as an objective function for the optimization problem.
The optimization problem with respect to the instances of the user selection and
power allocation optimization variables sk(n), Pk(n) is formulated as follows
Problem 2.1. EH Maximization:
maximizesk(n),Pk(n)
T∑
n=1
K∑
k=1
(1− sk(n))Ek(n) (2.3.1)
subject to C1 : sk(n) ∈ 0, 1,∀n, k,
C2 :K∑
k=1
sk(n)≤ 1,∀n,
C3 :1
T
T∑
n=1
K∑
k=1
Pk(n)sk(n)≤ Pav,
C4 :K∑
k=1
Pk(n)sk(n)≤ Pmax,∀n,
C5 :1
T
T∑
n=1
Ck(n)sk(n)≥ Creqk,∀k.
sk and Pk, k ∈ 1,2, . . . , K, are the vectors that represent the optimization variables,
i.e., user selection and power allocation variable, respectively. For notational simplicity,
in the following analysis and problem formulation, we use the instances of the variables
sk(n) and Pk(n), for each user k and time slot n, respectively. This also holds for other
variables dependent on the indices k and n, for users and time slots, respectively.
In the formulation of Problem 2.1, function Ek(n) is the power conversion function,
proposed in (2.2.2), modified corresponding to the parameters assumed in the system
model:
Ek(n) =
M
1+e−a(∑K
j=1 s j (n)Pj (n)hk(n)−b)− M
1+eab
1− M1+eab
. (2.3.2)
For notational simplicity, we rewrite (2.3.2) as
Ek(n) =
Ψk(n)−MΩ
1−Ω, where (2.3.3)
Ψk(n) =M
1+ e−a(∑K
j=1 s j(n)Pj(n)hk(n)−b), and (2.3.4)
Ω =1
1+ eab . (2.3.5)
16 Chapter 2. Resource Allocation Algorithm for a Practical EH Receiver Model
Ψk(n) is the standard logistic function with respect to the received power∑K
j=1 s j(n)Pj(n),transmitted to all the users selected for ID in a specific time slot n. In the following de-
velopment of the optimization problem, we use directly Ψk(n) from (2.3.4) to represent
the harvested power at a corresponding EH receiver, while ignoring the constant part Ω,
since it does not depend on the optimization variables. Without loss of generality, the
term (1− sk(n)) is included inside the objective function, i.e., in the exponential part of
Ψk(n). Thus, Problem 2.1 takes the following form:
Problem 2.2. EH Maximization:
maximizesk(n),Pk(n)
T∑
n=1
K∑
k=1
M
1+ e−a(PERk(n)hk(n)−b)
(2.3.6)
subject to C1,C2, C3,C4, C5.
Variable PERk(n) = (1− sk(n))(
∑Kj=1 s j(n)Pj(n)),∀n, k, represents the total power that is
received at EH receiver (ER) k at specific time slot n. The requirements of the system
are reflected in constraints C1–C5 in Problem 2.1 and 2.2. Constraints C1 and C2
are imposed to guarantee that in each time slot n at most one user is served by the
transmitter for information decoding. C3 imposes a constraint on the maximum of
average radiated power Pav and C4 shows the hardware limitations for the maximum
power Pmax that is allowed to be transmitted from the base station at each time slot.
Moreover, the QoS constraint is included into C5, where Ck(n) is the data rate for user k
and time slot n, defined in (2.1.2). C5 implies that the minimum required data per user
Creqkneeds to be achieved on average.
Problem 2.2 is a mixed non-convex and combinatorial problem. In order to exploit
standard convex optimization tools to efficiently solve the problem, Problem 2.2 needs
to be transformed into an equivalent1 problem with tractable structure. In the following,
we present the solution of the optimization problem.
2.4. Solution of the Optimization Problem
The non-convexity of the optimization Problem 2.2, arises from both the objective
function and the constraints. In particular, the objective function is a sum-of-ratios
function which does not enjoy convexity. Furthermore, the combinatorial nature is
imposed by the binary integer constraint C1 for the user selection variable. The first step
in solving the optimization problem is to transform the objective function.
1Two optimization problems are equivalent if the solution of one is readily obtained from the solution ofthe other problem. [58]
2.4. Solution of the Optimization Problem 17
2.4.1. Transformation of the Sum-of-ratios Objective Function
The sum-of-ratios optimization problem, which includes an objective function with a
sum of rational functions, is a non-convex problem that cannot be directly solved via
traditional optimization methods and optimization tools. Lately, several attempts for
solving this non-linear optimization problem have been presented in the literature. For
instance, the authors in [59] used the branch-and-bound method [60] along with in-
sights from recent developments in fractional programming and convex underestimators
theory in order to find the solution to a specific sum-of-ratio problem. However, their
methods result in relatively high computational complexity and only yield an approxi-
mation to the globally optimal solution. Another work in [61] focused on maximizing a
sum of sigmoidal functions subject to convex constraints, which resembles our problem
formulation. The authors proved that the defined problem is NP-hard and used the
branch-and-bound method to solve it, even thought they were also only able to give an
approximate solution to the problem. Along with the fact that these methods are not able
to obtain the globally optimal solution, the branch-and-bound method is of exponential
complexity, and may increase the computational time severely. Although there already
exist algorithms, such as the Dinkelbach method [62] or the Charnes-Cooper transforma-
tion [63], that solve the non-linear optimization problem for a single rational objective
function, they cannot be applied to the case with a sum-of-ratios objective function. Until
very recently, the algorithm introduced in [64], on the other hand, offered a solution to
the sum-of-ratios problem that is proven to achieve the global optimum. The crux of the
method is a transformation of the sum-of-ratios objective function into an equivalent
parametric convex optimization function, such that the globally optimal solution can be
successfully found through an iterative algorithm. The mentioned algorithm has been
initially used in several works [65]–[67], mostly for the optimization of different types
of system energy efficiency under different contexts. In [65], the authors focused on
the design of a resource allocation algorithm for jointly optimizing the energy efficiency
in downlink and uplink for networks with carrier aggregation. The authors in [66]used the algorithm for energy efficiency maximization framework in cognitive two-tier
networks. Moreover, a multi-cell, and multi-user precoding was designed in [67], with
the goal to maximize the weighted sum energy efficiency.
The main transformation, that the author in [64] proposed, converts the original
sum-of-ratio functions into a parametric subtractive form. This transformation allows
standard optimization tools to be further used and provides the ability to design an
efficient algorithm for achieving the globally optimal solution of the original sum-of-
ratios problem. The assumptions for this transformation require that the numerator
of the rational function of every summand is concave, and the denominator is convex
18 Chapter 2. Resource Allocation Algorithm for a Practical EH Receiver Model
and greater than zero. Thus, the transformed subtractive form is a concave function for
every summand in the case of maximization. We introduce the transformation of the
objective function from Problem 2.2, through the following theorem.
Theorem 1. Let s∗k(n), and P∗k (n) be the optimal solution to Problem 2.2, then there exist
two parameter vectors µ∗k and β∗k , k ∈ 1, 2, . . . , K. Furthermore, s∗k(n), and P∗k (n) are the
optimal solution to the following transformed optimization problem:
Problem 2.3. EH Maximization - Sum-of-Ratios Objective Function Transformation:
maximizesk(n),Pk(n)∈C
T∑
n=1
K∑
k=1
µ∗k(n)
M − β∗k(n)
1+ e−a(PERk(n)hk(n)−b)
. (2.4.1)
C is the feasible solution set of Problem 2.2 and PERk(n) = (1−sk(n))(
∑Kj=1 s j(n)Pj(n)),∀n, k.
In addition, the optimization variables s∗k(n), and P∗k (n) must satisfy the system of equa-
tions:
β∗k(n)
1+ e−a(P∗ERk(n)hk(n)−b)−M = 0, (2.4.2)
µ∗k(n)
1+ e−a(P∗ERk(n)hk(n)−b)− 1= 0,∀n, k. (2.4.3)
Proof. Please refer to Appendix A.1 for the proof.
As Theorem 1 suggested, there exists an optimization problem with an objective
function in subtractive form that is an equivalent problem to the sum-of-ratios Problem
2.2. More importantly, both optimization problems share the same optimal solution
and we can straightforwardly obtain the solution to the initial problem by solving the
transformed optimization problem [58], in the case when the transformed optimization
Problem 2.3 can be solved. Therefore, we can focus on the optimization problem with
the equivalent objective function in the rest of the thesis.
2.4.2. Iterative Algorithm for Maximization of Harvested Energy at
EH Receivers
In this subsection, we design a computationally efficient algorithm for achieving the
globally optimal solution of the resource allocation optimization Problem 2.2. To obtain
the solution for Problem 2.2, we adopt an equivalent objective function, such that the
resulting resource allocation policy satisfies the conditions in Theorem 1. The algorithm
has an iterative structure, consisting of two nested loops. Its structure is presented
in Table 2.1. The convergence to the globally optimum solution is guaranteed if the
transformed optimization Problem 2.3 can be solved in each iteration.
2.4. Solution of the Optimization Problem 19
Proof. For a proof of convergence, please refer to [64].
1: Initialize maximum number of iterations Imax, iteration index m = 0, µi, and βi,∀i ∈ 1, · · · , N, N = T K
2: repeat3: Solve the transformed inner loop convex optimization Problem 2.6 for given µm
iand βm
i and obtain the intermediate solution for si, Pvirtuali , and P ′i , ∀i
4: if (2.4.8) is satisfied then5: Convergence = true6: return optimal user selection and power allocation7: else8: Update µi and βi, ∀i, according to (2.4.9), and set m= m+ 19: Convergence = false
10: end if11: until Convergence = true or m= Imax
In each iteration of the inner loop, i.e., in lines 3–6 of the algorithm in Table 2.1, we
solve the following optimization problem for given µk(n) and βk(n), ∀n, k, and obtain
the optimal solution for the optimization variables sk(n), and Pk(n).
Problem 2.4. EH Maximization - Inner Loop Optimization Problem:
maximizesk(n),Pk(n)
T∑
n=1
K∑
k=1
µk(n)
M − βk(n)
1+ e−a(PERk(n)hk(n)−b)
(2.4.4)
subject to C1,C2, C3,C4, C5.
Although the objective function in Problem 2.4 is in subtractive form and is concave,
the transformed optimization problem is still non-convex due to the binary constraint
C1. To obtain a tractable problem formulation, we handle the binary constraint C1 from
Problem 2.4 in each iteration of the algorithm. For this purpose, we apply time-sharing
relaxation.
In particular, by following the approach in [68], we relax the user selection variable
sk(n) in constraint C1 of Problem 2.2 to take on real values between 0 and 1, i.e.,gC1: 0≤ sk(n)≤ 1,∀n, k. The user selection variable can now be interpreted as a time-
sharing factor for the K users during one time slot n. With the time-sharing relaxation,
the inner problem that we solve in each iteration takes the following form:
20 Chapter 2. Resource Allocation Algorithm for a Practical EH Receiver Model
Problem 2.5. EH Maximization - Time-sharing Relaxation:
maximizesk(n),P ′k(n)
T∑
n=1
K∑
k=1
µk(n)
M − βk(n)
1+ e−a((1−sk(n))∑K
j=1 P ′j (n)hk(n)−b)
(2.4.5)
subject to ÞC1 : 0≤ sk(n)≤ 1,∀n, k,
C2 :K∑
k=1
sk(n)≤ 1,∀n,
C3 :1
T
T∑
n=1
K∑
k=1
P ′k(n)≤ Pav,
C4 :K∑
k=1
P ′k(n)≤ Pmax,∀n,
C5 :1
T
T∑
n=1
sk(n) log2
1+P ′k(n)hk(n)
sk(n)σ2
≥ Creqk,∀k.
For facilitating the time-sharing, we introduce an auxiliary variable in Problem 2.5,
defined as P ′k(n) = Pk(n)sk(n), ∀n, k. The new optimization variable P ′k(n) represents
the actual transmitted power in the RF of the transmitter for user k at time slot n
under the time-sharing assumption. It also solves the problem with the coupling of
the optimization variables Pk(n), and sk(n), which is present in some of the constraints.
However, coupling of the variables is still present in the objective function after this
reformulation. Thus, we perform another variable change. In particular, we define the
variable Pvirtualk (n) = (1− sk(n))
∑Kk=1 P ′k(n), which represents the actual received power
at EH receiver k at a specific time slot n. After these changes, the inner loop optimization
problem is rewritten with respect to the optimization variables sk(n), P ′k(n), Pvirtualk (n):
2.4. Solution of the Optimization Problem 21
Problem 2.6. EH Maximization - Time-sharing Relaxation and Decoupling:
maximizesk(n),P ′k(n),P
virtualk (n)
T∑
n=1
K∑
k=1
µk(n)
M − βk(n)
1+ e−a(Pvirtualk (n)hk(n)−b)
(2.4.6)
subject to ÞC1 : 0≤ sk(n)≤ 1,∀n, k,
C2 :K∑
k=1
sk(n)≤ 1,∀n,
C3 :1
T
T∑
n=1
K∑
k=1
P ′k(n)≤ Pav,
C4 :K∑
k=1
P ′k(n)≤ Pmax,∀n,
C5 :1
T
T∑
n=1
sk(n) log2
1+P ′k(n)hk(n)
sk(n)σ2
≥ Creqk,∀k,
C6 : Pvirtualk (n)≤ (1− sk(n))Pmax,∀n, k,
C7 : Pvirtualk (n)≤
K∑
k=1
P ′k(n),∀n, k,
C8 : Pvirtualk (n)≥ 0,∀n, k.
Constraints C6–C8 are introduced due to the proposed transformation regarding the
auxiliary variable Pvirtualk (n). The constraints guarantee that the variable Pvirtual
k (n) retains
the physical meaning and is consistent with the original problem definition. This method
is referred to as the Big-M formulation in the literature [69].If the time-sharing relaxation is tight, then Problem 2.6 is equivalent to the original
optimization problem formulation in Problem 2.2. Now we study the tightness of the
time-sharing relaxation through the following theorem.
Theorem 2. The optimal solution of Problem 2.6 satisfies sk(n) ∈ 0, 1, ∀n, k. In particu-
lar, the user selection variable will still result in a solution at the boundaries of the relaxed
interval 0≤ sk(n)≤ 1.
Proof. Please refer to the Appendix A.2 for the proof.
Problem 2.6 represents a convex problem with convex constraints. Therefore, Problem
2.6 can be solved efficiently in each iteration of the algorithm in Table 2.1 using standard
convex optimization tools, e.g. CVX [70]. Now, we introduce the following proposition.
Proposition 2.1. The transformed Problem 2.6 is an equivalent transformation of the
original Problem 2.2. Thus, by solving Problem 2.6 in each iteration, we attain the optimal
solution to Problem 2.2.
22 Chapter 2. Resource Allocation Algorithm for a Practical EH Receiver Model
Proof. Please refer to Appendix A.3 for the proof.
The next step is to obtain an update for µk(n) and βk(n) to be used for solving the
inner loop optimization problem in the following iterations. This procedure represents
the outer loop of the algorithm. The algorithm is repeated until convergence to the
globally optimal solution is achieved. For notational simplicity, we introduce parameter
where i ∈ 1, · · · , N, and N = T K is the number of terms in the sum. In [64], it is
proven that the optimal solution ρ∗ = (µ∗,β∗) is achieved if and only if
ϕ(ρ) = [ϕ1, · · · ,ϕ2N] = 0 (2.4.8)
is satisfied. In the m-th iteration, we update ρ = (µ,β), in the following manner:
ρm+1 = ρm+ ζmqm, (2.4.9)
where qm = [ϕ′(ρ)]−1ϕ(ρ), and [·]−1 denotes the inverse of a matrix. Here, ϕ′(ρ) is
the Jacobian matrix of ϕ(ρ) [64]. Moreover, ζm is defined as the largest εl that satisfies:
‖ϕ(ρm+ εlqm)‖ ≤ (1−δεl)‖ϕ(ρm)‖, (2.4.10)
where l ∈ 1,2, · · · , εl ∈ (0,1), δ ∈ (0,1), and ‖·‖ denotes the Euclidean vector norm.
Equation (2.4.8) represents the convergence condition of the algorithm. For the update
of the respective variables, the modified Newton method is used, as shown in (2.4.9). If
ζm = 0, we have the well-known Newton method for the corresponding update. The
modified, or damped Newton method converges to the unique solution (µ∗i ,β∗i ),∀i,
while satisfying equations (2.4.2) and (2.4.3), with linear rate for any starting point
[64], [65]. The rate in the neighbourhood of the solution is quadratic, which follows
from the convergence analysis of the Newton method.
2.4.3. Dual Problem Formulation
In order to further investigate the structure of the solution, in this subsection we use
duality theory for solving the transformed optimization problem, cf. Problem 2.6. With
the corresponding transformations performed in the previous subsection, it can be
shown that Problem 2.6 is jointly concave with respect to the power allocation and user
selection variables. As a result, under some mild conditions, the solution of the dual
2.4. Solution of the Optimization Problem 23
problem is equivalent to the solution of the primal problem [58], i.e., strong duality
holds. Thus, we can use duality theory to obtain the solution. In order to do that, we
start with the formulation of the Lagrangian for Problem 2.6:
L (Pvirtualk (n), P ′k(n), sk(n),µk(n),βk(n),D) (2.4.11)
=T∑
n=1
K∑
k=1
µk(n)
M− βk(n)
1+ e−a(Pvirtualk (n)hk(n)−b)
−T∑
n=1
λ(n)
K∑
k=1
sk(n)− 1
−T∑
n=1
K∑
k=1
αk(n)
sk(n)− 1
+T∑
n=1
K∑
k=1
εk(n)sk(n)− γ 1
T
T∑
n=1
K∑
k=1
P ′k(n)− Pav
−T∑
n=1
δ(n)
K∑
k=1
P ′k(n)− Pmax
−K∑
k=1
ε(k)
Creqk−
1
T
T∑
n=1
sk(n) log2
1+P ′k(n)hk(n)
sk(n)σ2
−T∑
n=1
K∑
k=1
ζk(n)
Pvirtualk (n)−
1− sk(n)
Pmax
−T∑
n=1
K∑
k=1
ηk(n)
Pvirtualk (n)−
K∑
k=1
P ′k(n)
+T∑
n=1
K∑
k=1
θk(n)Pvirtualk (n),
where D = αk(n), λ(n), εk(n), γ, δ(n), ε(k), ζk(n), ηk(n), θk(n), ∀n, k, denotes
the set that contains all Lagrange multipliers. D is defined in order to simplify the
notation. In (2.4.11), αk(n), and εk(n), ∀n, k, are the Lagrange multipliers that account
for constraint C2, i.e., that only one user is chosen in one time slot n, along with λ(n),∀n,
that accounts for constraint C1. γ is the Lagrange multiplier related to the constraint
on the average radiated power implied by C3. δ(n) and ε(k), ∀n, k, account for the
maximum power transmitted from the base station during time slot n in C4 and the
minimum data rate requirements per user in C5, respectively. Furthermore, ζk(n), ηk(n),and θk(n), ∀n, k, are associated with the constraints C6–C8 related to the auxiliary
optimization variable Pvirtualk (n),∀n, k. The dual problem is given by:
minimizeD>0
maximizesk(n),P ′k(n),P
virtualk (n)
L (Pvirtualk (n), P ′k(n), sk(n),µk(n),βk(n),D). (2.4.12)
It can be shown that Problem 2.6 satisfies the Slater’s constraint qualification and strong
duality holds. Thus, from the Karush-Kuhn-Tucker (KKT) optimality conditions, the
gradient of the Lagrangian with respect to the elements of the optimization variables
24 Chapter 2. Resource Allocation Algorithm for a Practical EH Receiver Model
vanishes at the optimum point. First, we consider the derivatives of the Lagrangian with
respect to the instances of the optimization variables sk(n), P ′k(n), and Pvirtualk (n).
∂L∂ P ′k(n)
= ηk(n)− γ1
T−δ(n) +
1
T
ε(k)sk(n)ln2
hk(n)sk(n)σ2
1+P ′k(n)hk(n)
sk(n)σ2
, (2.4.13)
∂L∂ sk(n)
=ε(k)T ln2
ln(1+Pk(n)hk(n)
σ2 )−1
1+ Pk(n)hk(n)σ2
Pk(n)hk(n)σ2
+ εk(n)−λ(n)−αk(n)− ζk(n)Pmax, (2.4.14)
∂L∂ Pvirtual
k (n)= θk(n)− ζk(n)−ηk(n) + auk(n)βk(n)hk(n)e
−a(Pvirtualk (n)hk(n)−b). (2.4.15)
By exploiting the fact that the derivative of the Lagrangian with respect to the opti-
mization variable P ′k(n) vanishes at the optimum point, from (2.4.13), we obtain the
following
P ′k(n) = sk(n)Pk(n)
= sk(n)
ε(k)ln 2(γ+ Tδ(n)− Tηk(n))
−σ2
hk(n)
+
. (2.4.16)
From (2.4.15), taking the derivative of the Lagrangian with respect to Pvirtualk (n) yields:
Pvirtualk (n) =
b
hk(n)−
1
ahk(n)lnζk(n) +ηk(n)− θk(n)
auk(n)βk(n)hk(n)
+
. (2.4.17)
The structure of the solution at the optimum point can be observed from the results
presented above. In particular, it can be observed from (2.4.16) that the power allo-
cation in the system follows the water-filling solution. The dual variables show the
costs for realizing the specific power allocation. Namely, in (2.4.16), we can observe
that P ′k(n) as an auxiliary variable is defined as the coupling between the true power
allocation variable Pk(n) and the user selection variable. Regarding the power allocation
Pk(n), which follows the water-filling policy, we can notice a ratio between the dual
variables. Specifically, we can observe the dual variable dedicated to the rate constraint
in the numerator and the dual variables connected to the power constraints from the
transformed optimization problem in the denominator. The Lagrange multipliers ε(k),γ, δ(n), and ηk(n), ∀n, k make sure that the transmitter transmits with a sufficient
amount of power to fulfill the data rate requirements, while satisfying the average and
maximum power constrains. Moreover, in equation (2.4.16) we can observe a part that
is inverse to the channel value hk(n) for user k and time slot n, which follows the form of
2.5. Results 25
water-filling, i.e., users with better channel conditions at a specific time slot are allocated
more power.
2.5. Results
In this section, we present simulation results to illustrate the system performance of
the proposed resource allocation algorithm with respect to the non-linear practical EH
receiver model, proposed in Section 2.2. The important parameters adopted in the
simulation are summarized in Table 2.2. Regarding the noise variances at the receiver
Table 2.2.: Simulation ParametersCarrier center frequency 915 MHzBandwidth B = 200 kHzReceiver antenna noise power σ2 =−111.9 dBmNumber of users K 10 / 15Number of time slots T 100Transmit antenna gain 18 dBiReceiver antenna gain 0 dBiPath loss exponent 2Rician factor 0 dBReference and maximum service distance (w.r.t. Pmax) 10 metersReference and maximum service distance (w.r.t. distance) 10 - 30 metersConstraint on average radiated power Pav 0.2Pmax
Maximum transmit power Pmax (w.r.t. Pmax) 36− 46 dBmMaximum transmit power Pmax (w.r.t. distance) 46 dBmMaximum harvested DC power for rectifying circuit - M 20 mWEH circuit parameter - a 1500EH circuit parameter - b 0.0022Minimum required data rate per user Creqk
Creq1= 0.5 bit/s/Hz,
Creqi= 0bit/s/Hz, i = 1 . . . K
antennas, we assume that they are identical at the information receiver and the EH
receivers. The value for the corresponding noise power includes the effect of thermal
noise at a temperature of 290 Kelvin and the processing noise. The results are simulated
for 10 and 15 users in the system over 100 time slots for computing the total average
harvested power. We assume the path loss model defined in [71], with a path loss
exponent of 2. The multipath fading coefficients are modelled as independent and
identically distributed Rician fading. The impedance of the antennas at the receivers is
assumed to have perfect matching to the rectifying circuit such that there is no additional
power losses. For the non-linear EH receiver model, i.e., the parameters for the non-
linear EH circuits, we set M = 20 mW which corresponds to the maximum harvested
26 Chapter 2. Resource Allocation Algorithm for a Practical EH Receiver Model